Do models train better if the labelling information is more specific (or dense)?
It depends on what is your ultimate goal. If your goal is to simply classify the object in the image, having more complex output won't help. Simpler output representation yields better result. If your goal is to detect the bounding box, output the bounding box. There is no need for a more complex output feature. If you use a segmentation method for bounding box detection, it is more prone to error because of it's excess output features.

Assume you are given a gradr 6 math test. If you do the questions using grade 12 maths knowledge and do it with calculus and stuff to make your calculations seems more complex, will you have a higher mark than doing it the normal way? No! The marks is the same or even less due to higher chance of error in doing complex calculations.

In short, higher complexity on your labels won't help your task if it is a simple task. Hope this would help you and have a nice day!